Presentation on theme: "The Asymptotic Ray Theory"— Presentation transcript:
1 The Asymptotic Ray Theory Green functions for the asymptotic high frequency approximationIn order to compute the seismic waves radiated by an earthquake, it is necessary to appropriately compute the Green Functions to model the wave propagation in a given crustal medium.The accuracy for computing Green Functions depends on the detailed knowledge of the Earth crust. It is evident that computing high frequency(f > 1 Hz) Green Functions requires the knowledge of the complex 3-D structure of the crust.It is assumed that the high frequency component of seismic waves propagates along particular trajectories called rays; therefore, it relies on the substitution of the wave front with its normal, which is the seismic ray.
2 The Eikonal equation EQUATION OF MOTION TENTATIVE SOLUTION SOLUTION APPROXIMATIONWe have derived the first important equation in the framework of ray theory: the travel time of a seismic wave follows the Fermat’s principleIn optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be traversed in the least time.
3 The ray equation wave front travel time ray parametric equation The second important equation is the equation of the seismic ray; that is the equation that allows the identification of the trajectory followed by a seismic wave to propagate from x to x.travel timeray parametric equationtentative solution
4 The ray equationThis equation is relatively simple to solve, but it has only a kinematic meaning. In other words, by solving equation given a wave velocity profile it is possible to find the path followed by the seismic ray.is parallel toc is constantconstant along the rayp is the ray parameter
5 Plane waves in 3D homogenous medium v = w / k = l / TPlane waves in 3D homogenous medium
6 The ray coordinate system In order to study the ray-tracing problem, it is more convenient to avoid the Cartesian coordinate system and to use a more appropriate local reference framework.
7 The ray coordinate system In order to study the ray-tracing problem, it is more convenient to avoid the Cartesian coordinate system and to use a more appropriate local reference framework.
8 The ray coordinate system In order to study the ray-tracing problem, it is more convenient to avoid the Cartesian coordinate system and to use a more appropriate local reference framework.
9 The slowness vectorPropertiesComponents in 2DRay equation
10 Depth dependent velocity model ABDepth dependent velocity model⌫★Cartesian Coordinates
11 Seismic Waves in a spherical Earth Ray parameter in a spherical coordinate systemiv(r)r being the distance from the centre of symmetry
13 THE GEOMETRICAL SPREADING in a homogenous medium (that is, constant body wave velocity) the rays are lines and the amplitudes scale with a geometrical factor being proportional tothe radiation pattern of P-wavesradiated in different directionstime function related to the body forceat the source giving a displacement pulsein the longitudinal directionCartesian3 coordinate systemthe general form of the geometricalspreading.local mobilefor a homogeneous caseLocal spherical
14 THE GEOMETRICAL SPREADING Geometrical spreading of four rays at two different values of travel time (to, t)THE GEOMETRICAL SPREADINGds(to) and ds(t) are the twoelementary surfaces describingthe section of the ray tube on thewave front at different times.The geometrical spreading factor in inhomogeneous media describes the focusing and defocusing of seismic rays. In other words, the geometrical spreading can be seen as the density of arriving rays; high amplitudes are expected where rays are concentrated and low amplitudes where rays are sparse.The focusing or defocusing of the rays can be estimatedby measuring the areal section on the wave front atdifferent times defined by four rays limiting an elementaryray tube.Each elementary area at a given time is proportional tothe solid angle defining the ray tube at the source , butthe size of the elementary area varies along the ray tube.
15 displacement field radiated by a double couple homogeneousThe analytical expression of the geometrical spreading factor depends on the general properties of the orthogonal curvilinear coordinates adopted to define the local framework systemspherically symmetric
16 Seismic wave Energy The strain energy density This relation comes out from considering that the mechanical work ( W) is a function of strain components and is equal toThe kinetic energyP waveS waveFor a steady state plane wave incidenton a boundary between two homogeneous half spaces the energy flux leaving the boundary must equal that in the incident wave.That is there is no trapped energy at the interface
17 Reflection and transmission coefficients for seismic waves The internal structure of solid Earth is characterized by the distribution of physical properties that affect seismic wave propagation and, therefore, can be studied by analyzing seismic waves.For seismological purposes, this is done by assigning the distribution of elastic properties and density or equivalently of seismic wave velocity and density.
18 Two homogenous medium in contact Snell Law v1v2i1 = i3i3 = reflection anglei1 = incidence angleREFLECTIONi2 = refraction angleREFRACTIONRay parameterAmplitudes are distributed between different waves (reflection and refraction coefficients)sin(i1)v1sin(i2)v2= p =
19 THE SNELL LAWSuppose that a plane P-wave is travelling with horizontal slowness in a direction forming and angle with the normal to the interface.A P-wave incident from medium 1 generates reflected and transmitted P-waves. In addition, part of this P-wave is converted into a reflected SV-wave and a transmitted SV wave.
21 Two homogenous medium in contact SH waves x2x1x3sin(j1)b1sin(j2)b2=
22 Two homogenous medium in contact SV waves Incident SV waveReflected & refracted SVi2sen(i2)= (b2/b1) sen(i1)a2, b2SVa1, b1SVi1a2<a1b2<b1SV
23 Two homogenous medium in contact SV waves Reflected & Refracted P wavessen(i2)= (a2/b1) sen(i1)a2, b2i2PPa1, b1i3i1a2<a1b2<b1sen(i3)= (a1/b1) sen(i1)SV
24 Critical Incidence If the second medium has a higher velocity, Because in any medium P-waves travel faster than S-waves ( ), Snell’s law requires that Moreover, the angle of incidence for refracted P-wave is related to that of the incident P-wave byS-wave reflectedIf the second medium has a higher velocity,the transmitted P-wave is farther from the vertical(i.e., more horizontal) than the incident wave.As the incidence angle increases, the transmitted waves approach the direction of the horizontal interface. The incidence angle reaches a value
25 Critical Incidence incident P-waves on a faster medium incident SH wave only generates reflectedand transmitted SH waves
26 The ray parameter again! It is useful to remind that the ray parameter ishorizontal wavenumberapparent velocityFor a P-wave the slowness vector is given by
30 Seismic Wave propagation. As seismic waves travel through Earth, they interact with the internal structure of the planet and:Refract – bend / change directionReflect – bounce off of a boundary (echo)Disperse – spread out in time (seismogram gets longer)Attenuate – decay of wave amplitudeDiffract – non-geometric “leaking” of wave energyScatter – multiple bouncing around
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